Scalar: An element belonging to either one of the fields GF (p) or GF (2k) is called a scalar.
Scalars are named with lowercase letters: r, s, t, etc.
bScalar Addition: Two or more scalar can be added to obtain another scalar. In the GF (p) case, this is the ordinary integer addition modulo p. When GF (2k) is used, this is equivalent to polynomial addition modulo an irreducible polynomial of degreek, generating the field GF (2k). We will denote the scalar addition of r and s giving the result e by e = r + s. Scalar Multiplication: Two or more scalar can be multiplied to obtain another scalar. In the GF (p) case, this is the ordinary integer multiplication modulo p. When GF (2k) is used, this is equivalent to polynomial multiplication modulo an irreducible polynomial of degree k, generating the field GF (2k). We will denote the scalar multiplication of r and s giving the result e by e = rs.
• Scalar Inversion: The multiplicative inverse of an element a in GF (p) or GF (2k)is denoted as a−1 which is the number with the property aa−1 = 1. It is often computed using the Fermat’s method or the extended Euclidean algorithm.
• Point: An ordered pair of scalars satisfying theelliptic curve equationis called apoint. Capital letters are used to denote these elements: P ,Q, etc. We will also denote a point P using its coordinates P = (x, y), where x and y belong to the field. Furthermore, the x and y coordinates of P will be denoted by P.x or P.y, respectively.
&bPoint Addition: There is a method to obtain a third point R on the curve given two points P and Q, using a set of rules. This is called an elliptic curve point addition. We will use the symbol ‘+’ to denote the elliptic curve addition R = P + Q. This should not be confused with scalar addition.
2.6 Elliptic Curve Group: The set of the solutions of the elliptic curve equation together with a special point called point-at-infinity form an additive group if the point addition operation defined above is taken as the group operation.
Point Multiplication: The multiplication of an elliptic curve point P by an integer e will be denoted by e × P . It is equivalent to adding P to itself e times, whichyields another point on the curve.
In addition to the above elliptic curve cryptographic primitives, we often need a one-way hash (message digest) function which is defined below:
1.6 Message Digest Function: A message digest function compresses a long message into a short value which is usually 128 or 160 bits long. Two widely used and standardized hash functions are MD5 and SHA. We will denote the message digest of a message.
M byH(M). The signature functions takeH(M) as an input for efficiency reasons.The hash of the concatenation of two messages M1 and M2 is denoted as H(M1, M2).
3.4 RELATED WORK ON ELLIPTIC CURVE CRYPTOSYSTEM
The dominant security protocol for handling security over the internet has been Secure Socket Layer (SSL) protocol which has its technology based on public key cryptography (PKC). In PKC, Rivest, Shamir and Adleman (RSA), Digital Signature authentication (DSA) and Diffie- Hellman, protocols are used traditionally. Forecasters predict more than a billion wireless users by 2005. As the wireless industry explodes, it faces a growing need for security. Applications in sectors of the economy such as healthcare, financial services, and government depend on the underlying security already available in the wired computing environment. Both for secure (authenticated, private) Web transactions and for secure (signed, encrypted) messaging, a full and efficient public key infrastructure is needed.
Three basic choices for public key systems are available for these applications:
➢ RSA
➢ Diffie-Hellman (DH) or Digital Signature Algorithm (DSA) modulo a prime p
➢ Elliptic Curve Diffie-Hellman (ECDH) or Elliptic Curve Digital Signature Algorithm (ECDSA)
3.4.1. RSA (Rivest, Shamir, and Adleman)
RSA is a system that was published in 1978 by Rivest, Shamir, and Adleman, based on the difficulty of factoring large integers. Whitfield Diffie and Martin Hellman proposed the public key system now called Diffie-Hellman Key Exchange in 1976. DH is key agreement and DSA is signa-ture, and they are not directly interchangeable, although they can be combined to do authenticated key agreement. Both the key exchange and digital signature algorithm are based on the difficulty of solving the discrete logarithm problem in the multiplicative group of integers modulo a prime p. Elliptic curve groups were proposed in 1985 as a substitute for the
multiplicative groups modulo p in either the DH or DSA protocols.
